Related papers: Generalized relations between arithmetic functions

The classical Mertens' formula states that $ \prod_{p\le N}\big(1-\frac1p)^{-1}\sim e^\gamma\log N, $ where the product is over all primes $p$ less than or equal to $N$, and $\gamma$ is the Euler-Mascheroni constant. By the Euler product…

A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side…

Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ \zeta(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $\zeta(z)$, the generalized Stieltjes constants $\gamma_k(q)$, the…

We obtain asymptotic formulas for the sums $\sum_{n_1,\ldots,n_k\le x} f((n_1,\ldots,n_k))$ and $ \sum_{n_1,\ldots,n_k\le x} f([n_1,\ldots,n_k])$ involving the gcd and lcm of the integers $n_1,\ldots,n_k$, where $f$ belongs to certain…

Using basic hypergeometric functions and partial fraction decomposition we give a new kind of generalization of identities due to Uchimura, Dilcher, Van Hamme, Prodinger, and Chen-Fu related to divisor functions. An identity relating…

We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series…

An important unsolved question in number theory is the Lehmer's totient problem that asks whether there exists any composite number $n$ such that $\varphi(n)\mid n-1$, where $\varphi$ is the Euler's totient function. It is known that if any…

We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order (p-primary) orbifold Euler characteristic of symmetric…

We construct a generalization of the multiplicative product of distributions presented by L. H\"ormander in [L. H\"ormander, {\it The analysis of linear partial differential operators I} (Springer-Verlag, 1983)]. The new product is defined…

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…

Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the number of preimages of Euler's totient function $\phi$ and the sum-of-divisors functions $\sigma$. In this paper, we will extend the upper bound to the number of…

Recently, by the Riordan's identity related to tree enumerations, \begin{eqnarray*} \sum_{k=0}^{n}\binom{n}{k}(k+1)!(n+1)^{n-k} &=& (n+1)^{n+1}, \end{eqnarray*} Sun and Xu derived another analogous one, \begin{eqnarray*}…

In the quantum theory, using the notion of partial supersymmetry, in which some, but not all, operators have superpartners we derive the Euler theorem in partition theory. The paraferminic partition function gives another identity in…

We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove…

We give a representation of the classical Riemann $\zeta$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen…

The Euler's totient function $ \varphi(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ \varphi(n)\mid n-1…

The partial fraction expansion of coth($\pi$z), due to Euler, is generalized to power series having for coefficients the Riemann zeta function evaluated at certain arithmetic sequences. A further generalization using arbitrary Dirichlet…

We deduce asymptotic formulas for the sums $\sum_{n_1,\ldots,n_r\le x} f(n_1\cdots n_r)$ and $\sum_{n_1,\ldots,n_r\le x} f([n_1\cdots n_r])$, where $r\ge 2$ is a fixed integer, $[n_1,\ldots,n_r]$ stands for the least common multiple of the…

The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and $p$-adic integral (the Volkenborn integral). By using these…

Erd\H{o}s first conjectured that infinitely often we have $\varphi(n) = \sigma(m)$, where $\varphi$ is the Euler totient function and $\sigma$ is the sum of divisor function. This was proven true by Ford, Luca and Pomerance in 2010. We ask…